how to isolate and texture a certain area of a mesh
If you don't want to use any image texture you can do it with Vertex Paint:


Switch to Vertex Paint mode, open the Tool panel, choose a black color, select the Blend > Mix mode, paint around the eyes. The vertex paint only uses the vertices, so its smoothness and precision will depend on your mesh density:





As you can see it has created a Vertex Colors group:





In your Shade Editor, create a Input > Attribute node. In the Name field, type the name of the vertex color group you've just created. Now you can use it as a factor in a Mix Shader: